Logic As A Tool For Building Theories

Transcription

1 Logic As A Tool For Building Theories Samson Abramsky Oxford University Computing Laboratory Logic For Building Theories Practice-Based PoL: Amsterdam / 29

2 Overview The Plan Logic For Building Theories

3 Overview Overview The Plan Practice-based Philosophy of Logic? Logic For Building Theories Practice-Based PoL: Amsterdam / 29

4 Overview Overview The Plan Practice-based Philosophy of Logic? What is the practice? Logic For Building Theories Practice-Based PoL: Amsterdam / 29

5 Overview Overview The Plan Practice-based Philosophy of Logic? What is the practice? Huge effect of Computer Science on Logic over the past 5 decades: new ways of using logic new attitudes to logic new questions new methods Logic For Building Theories Practice-Based PoL: Amsterdam / 29

6 Overview Overview The Plan Practice-based Philosophy of Logic? What is the practice? Huge effect of Computer Science on Logic over the past 5 decades: new ways of using logic new attitudes to logic new questions new methods Hence new perspective on the question: What logic is and should be! Logic For Building Theories Practice-Based PoL: Amsterdam / 29

7 The Plan Overview The Plan Before (and while) trying to extract general points, some case studies: Logic For Building Theories Practice-Based PoL: Amsterdam / 29

8 The Plan Overview The Plan Before (and while) trying to extract general points, some case studies: modal and temporal logic: verification and model-checking Logic For Building Theories Practice-Based PoL: Amsterdam / 29

9 The Plan Overview The Plan Before (and while) trying to extract general points, some case studies: modal and temporal logic: verification and model-checking Logic For Building Theories Practice-Based PoL: Amsterdam / 29

10 The Plan Overview The Plan Before (and while) trying to extract general points, some case studies: modal and temporal logic: verification and model-checking coalgebra Logic For Building Theories Practice-Based PoL: Amsterdam / 29

11 The Plan Overview The Plan Before (and while) trying to extract general points, some case studies: modal and temporal logic: verification and model-checking coalgebra Aims: not to put forward any philosophical theses, but to provide some materials and raise some questions. Logic For Building Theories Practice-Based PoL: Amsterdam / 29

12 Changing Perspectives Some Issues Some Remarks and Questions for P-B PoL The Next 700 Problem and Temporal Logic Logic For Building Theories

13 Changing Perspectives Logic For Building Theories Practice-Based PoL: Amsterdam / 29

14 Changing Perspectives From philosophical logic to computer-assisted verification. From metaphysics to (not just potentially but actually) applied mathematics. Logic For Building Theories Practice-Based PoL: Amsterdam / 29

15 Changing Perspectives From philosophical logic to computer-assisted verification. From metaphysics to (not just potentially but actually) applied mathematics. From the sacred to the profane. From logic as Guardian of The Truth to logic out in the world, to be used as a tool for understanding many aspects of our world. Logic For Building Theories Practice-Based PoL: Amsterdam / 29

16 Changing Perspectives From philosophical logic to computer-assisted verification. From metaphysics to (not just potentially but actually) applied mathematics. From the sacred to the profane. From logic as Guardian of The Truth to logic out in the world, to be used as a tool for understanding many aspects of our world. More concretely: from possible worlds and accessibility to states and transitions. Systems of states evolving under discrete transitions turn up in a huge variety of situations. (Communications protocols, hardware circuits, software, nowadays biological and physical systems... ). Modal and temporal logics are canonical formalisms for expressing and reasoning about properties of such systems. Logic For Building Theories Practice-Based PoL: Amsterdam / 29

17 Changing Perspectives From philosophical logic to computer-assisted verification. From metaphysics to (not just potentially but actually) applied mathematics. From the sacred to the profane. From logic as Guardian of The Truth to logic out in the world, to be used as a tool for understanding many aspects of our world. More concretely: from possible worlds and accessibility to states and transitions. Systems of states evolving under discrete transitions turn up in a huge variety of situations. (Communications protocols, hardware circuits, software, nowadays biological and physical systems... ). Modal and temporal logics are canonical formalisms for expressing and reasoning about properties of such systems. A transition system: a b 1 2 c Logic For Building Theories Practice-Based PoL: Amsterdam / 29

18 Some Issues Logic For Building Theories Practice-Based PoL: Amsterdam / 29

19 Some Issues Note the reverse engineering here. Historically, formal systems of modal logic were developed to study notions of necessity. Then Kripke semantics was developed to shed light on these formal systems. Now we think of the structures as the naturally occurring objects of study, the logics as tools for reasoning about them. Logic For Building Theories Practice-Based PoL: Amsterdam / 29

20 Some Issues Note the reverse engineering here. Historically, formal systems of modal logic were developed to study notions of necessity. Then Kripke semantics was developed to shed light on these formal systems. Now we think of the structures as the naturally occurring objects of study, the logics as tools for reasoning about them. New questions: algorithmic feasibility. New methods: Logic For Building Theories Practice-Based PoL: Amsterdam / 29

21 Some Issues Note the reverse engineering here. Historically, formal systems of modal logic were developed to study notions of necessity. Then Kripke semantics was developed to shed light on these formal systems. Now we think of the structures as the naturally occurring objects of study, the logics as tools for reasoning about them. New questions: algorithmic feasibility. New methods: model-checking. Given a system description (a transition system) S, and a property φ we wish the system to satisfy, check if S = φ. This has become an enormously influential paradigm over the past 25 years. Much of the real value lies in cases where the property is not satisfied, and we get a trace which can lead us to the bug. Logic For Building Theories Practice-Based PoL: Amsterdam / 29

22 Some Issues Note the reverse engineering here. Historically, formal systems of modal logic were developed to study notions of necessity. Then Kripke semantics was developed to shed light on these formal systems. Now we think of the structures as the naturally occurring objects of study, the logics as tools for reasoning about them. New questions: algorithmic feasibility. New methods: model-checking. Given a system description (a transition system) S, and a property φ we wish the system to satisfy, check if S = φ. This has become an enormously influential paradigm over the past 25 years. Much of the real value lies in cases where the property is not satisfied, and we get a trace which can lead us to the bug. The automata-theoretical paradigm. Encode formulas as automata, reduce satisfaction to language inclusion, ultimately to graph reachability. Logic For Building Theories Practice-Based PoL: Amsterdam / 29

23 Some Issues Note the reverse engineering here. Historically, formal systems of modal logic were developed to study notions of necessity. Then Kripke semantics was developed to shed light on these formal systems. Now we think of the structures as the naturally occurring objects of study, the logics as tools for reasoning about them. New questions: algorithmic feasibility. New methods: model-checking. Given a system description (a transition system) S, and a property φ we wish the system to satisfy, check if S = φ. This has become an enormously influential paradigm over the past 25 years. Much of the real value lies in cases where the property is not satisfied, and we get a trace which can lead us to the bug. The automata-theoretical paradigm. Encode formulas as automata, reduce satisfaction to language inclusion, ultimately to graph reachability. Huge expansion to cover real-time, probabilistic and hybrid systems, and of applications to include biological systems, security, networks, agent-based modelling, control systems etc. Logic For Building Theories Practice-Based PoL: Amsterdam / 29

24 Some Remarks and Questions for P-B PoL Changing Perspectives Some Issues Some Remarks and Questions for P-B PoL The Next 700 Problem Logic For Building Theories Practice-Based PoL: Amsterdam / 29

25 Some Remarks and Questions for P-B PoL Changing Perspectives Some Issues Some Remarks and Questions for P-B PoL The Next 700 Problem The attitude that is is just grubby engineering engineers in overalls infringing on the sacred groves just won t wash. Logic For Building Theories Practice-Based PoL: Amsterdam / 29

26 Some Remarks and Questions for P-B PoL Changing Perspectives Some Issues Some Remarks and Questions for P-B PoL The Next 700 Problem The attitude that is is just grubby engineering engineers in overalls infringing on the sacred groves just won t wash. (Cf. 17th century vis-à-vis the Greeks. McCarthy!). Logic For Building Theories Practice-Based PoL: Amsterdam / 29

27 Some Remarks and Questions for P-B PoL Changing Perspectives Some Issues Some Remarks and Questions for P-B PoL The Next 700 Problem The attitude that is is just grubby engineering engineers in overalls infringing on the sacred groves just won t wash. (Cf. 17th century vis-à-vis the Greeks. McCarthy!). Concrete interpretations grounded in tangible applications which have an independent existence for their own reasons transform the possibilities and give the subject new depth and new energies. Logic For Building Theories Practice-Based PoL: Amsterdam / 29

28 Some Remarks and Questions for P-B PoL Changing Perspectives Some Issues Some Remarks and Questions for P-B PoL The Next 700 Problem The attitude that is is just grubby engineering engineers in overalls infringing on the sacred groves just won t wash. (Cf. 17th century vis-à-vis the Greeks. McCarthy!). Concrete interpretations grounded in tangible applications which have an independent existence for their own reasons transform the possibilities and give the subject new depth and new energies. There are passionate methodological debates within this applied field, e.g. linear time vs. branching time, which are fertile ground for conceptual and philosophical analysis. Feasibility becomes a major new criterion, and approximate answers must be considered. Such issues are already deeply embedded in physics, but rarely studied philosophically they should be! Logic For Building Theories Practice-Based PoL: Amsterdam / 29

29 Some Remarks and Questions for P-B PoL Changing Perspectives Some Issues Some Remarks and Questions for P-B PoL The Next 700 Problem The attitude that is is just grubby engineering engineers in overalls infringing on the sacred groves just won t wash. (Cf. 17th century vis-à-vis the Greeks. McCarthy!). Concrete interpretations grounded in tangible applications which have an independent existence for their own reasons transform the possibilities and give the subject new depth and new energies. There are passionate methodological debates within this applied field, e.g. linear time vs. branching time, which are fertile ground for conceptual and philosophical analysis. Feasibility becomes a major new criterion, and approximate answers must be considered. Such issues are already deeply embedded in physics, but rarely studied philosophically they should be! A deep conceptual issue of logic in CS: the next problem. Logic For Building Theories Practice-Based PoL: Amsterdam / 29

30 The Next 700 Problem Logic For Building Theories Practice-Based PoL: Amsterdam / 29

31 The Next 700 Problem After Peter Landin. The next 700 Programming Languages (in 1966!) Logic For Building Theories Practice-Based PoL: Amsterdam / 29

32 The Next 700 Problem After Peter Landin. The next 700 Programming Languages (in 1966!) A profusion of possibilities, in e.g. programming languages type systems process calculi behavioural equivalences logics Logic For Building Theories Practice-Based PoL: Amsterdam / 29

33 The Next 700 Problem After Peter Landin. The next 700 Programming Languages (in 1966!) A profusion of possibilities, in e.g. programming languages type systems process calculi behavioural equivalences logics Is this profusion a scandal of our subject? Or are the underlying paradigms and templates, the methodological toolkits, sufficient providers of unity? Logic For Building Theories Practice-Based PoL: Amsterdam / 29

34 The Next 700 Problem After Peter Landin. The next 700 Programming Languages (in 1966!) A profusion of possibilities, in e.g. programming languages type systems process calculi behavioural equivalences logics Is this profusion a scandal of our subject? Or are the underlying paradigms and templates, the methodological toolkits, sufficient providers of unity? The jury is still out... Logic For Building Theories Practice-Based PoL: Amsterdam / 29

35 The Next 700 Problem After Peter Landin. The next 700 Programming Languages (in 1966!) A profusion of possibilities, in e.g. programming languages type systems process calculi behavioural equivalences logics Is this profusion a scandal of our subject? Or are the underlying paradigms and templates, the methodological toolkits, sufficient providers of unity? The jury is still out... Cf. André Weil: he compared finding the right definitions in algebraic number theory which was like carving adamantine rock to making definitions in the theory of uniform spaces (which he founded), which was like sculpting with snow. Logic For Building Theories Practice-Based PoL: Amsterdam / 29

36 The Remarks Types and Curry-Howard Correspondence Developments Domain Theory Some Remarks and Questions for P-B PoL Logic For Building Theories

37 The : a pure calculus of functions. The Remarks Types and Curry-Howard Correspondence Developments Domain Theory Some Remarks and Questions for P-B PoL Logic For Building Theories Practice-Based PoL: Amsterdam / 29

38 The The Remarks Types and Curry-Howard Correspondence Developments Domain Theory Some Remarks and Questions for P-B PoL : a pure calculus of functions. Variables x, y, z,... Terms t ::= x tu }{{} application λx. t }{{} abstraction Logic For Building Theories Practice-Based PoL: Amsterdam / 29

39 The The Remarks Types and Curry-Howard Correspondence Developments Domain Theory Some Remarks and Questions for P-B PoL : a pure calculus of functions. Variables x, y, z,... Terms t ::= x tu }{{} application λx. t }{{} abstraction The basic equation governing this calculus is β-conversion: E.g. (λx. t)u = t[u/x] (λf. λx. f(fx))(λx. x + 1)0 = 2. Logic For Building Theories Practice-Based PoL: Amsterdam / 29

40 The The Remarks Types and Curry-Howard Correspondence Developments Domain Theory Some Remarks and Questions for P-B PoL : a pure calculus of functions. Variables x, y, z,... Terms t ::= x tu }{{} application λx. t }{{} abstraction The basic equation governing this calculus is β-conversion: E.g. (λx. t)u = t[u/x] (λf. λx. f(fx))(λx. x + 1)0 = 2. By orienting this equation, we get a dynamics β-reduction (λx. t)u t[u/x] Logic For Building Theories Practice-Based PoL: Amsterdam / 29

41 Remarks Logic For Building Theories Practice-Based PoL: Amsterdam / 29

42 Remarks This calculus, encapsulated in one slide, is incredibly rich. Logic For Building Theories Practice-Based PoL: Amsterdam / 29

43 Remarks This calculus, encapsulated in one slide, is incredibly rich. A universal model of computation incomparably more wieldy than Turing machines. Logic For Building Theories Practice-Based PoL: Amsterdam / 29

44 Remarks This calculus, encapsulated in one slide, is incredibly rich. A universal model of computation incomparably more wieldy than Turing machines. (Caveats: Church s thesis, resources). Logic For Building Theories Practice-Based PoL: Amsterdam / 29

45 Remarks This calculus, encapsulated in one slide, is incredibly rich. A universal model of computation incomparably more wieldy than Turing machines. (Caveats: Church s thesis, resources). Indeed, in sugared form the basis of all contemporary functional programming languages (e.g. ML, Haskell). Logic For Building Theories Practice-Based PoL: Amsterdam / 29

46 Remarks This calculus, encapsulated in one slide, is incredibly rich. A universal model of computation incomparably more wieldy than Turing machines. (Caveats: Church s thesis, resources). Indeed, in sugared form the basis of all contemporary functional programming languages (e.g. ML, Haskell). Kleene translated the basic results of recursion theory out of lambda calculus into the familiar φ n form. Logic For Building Theories Practice-Based PoL: Amsterdam / 29

47 Remarks This calculus, encapsulated in one slide, is incredibly rich. A universal model of computation incomparably more wieldy than Turing machines. (Caveats: Church s thesis, resources). Indeed, in sugared form the basis of all contemporary functional programming languages (e.g. ML, Haskell). Kleene translated the basic results of recursion theory out of lambda calculus into the familiar φ n form. The untyped calculus allows e.g. terms like ω λx. xx self-application. Logic For Building Theories Practice-Based PoL: Amsterdam / 29

48 Remarks This calculus, encapsulated in one slide, is incredibly rich. A universal model of computation incomparably more wieldy than Turing machines. (Caveats: Church s thesis, resources). Indeed, in sugared form the basis of all contemporary functional programming languages (e.g. ML, Haskell). Kleene translated the basic results of recursion theory out of lambda calculus into the familiar φ n form. The untyped calculus allows e.g. terms like ω λx. xx self-application. Hence also Ω ωω, which diverges: Ω Ω Logic For Building Theories Practice-Based PoL: Amsterdam / 29

49 Remarks This calculus, encapsulated in one slide, is incredibly rich. A universal model of computation incomparably more wieldy than Turing machines. (Caveats: Church s thesis, resources). Indeed, in sugared form the basis of all contemporary functional programming languages (e.g. ML, Haskell). Kleene translated the basic results of recursion theory out of lambda calculus into the familiar φ n form. The untyped calculus allows e.g. terms like ω λx. xx self-application. Hence also Ω ωω, which diverges: Ω Ω Also, Y λf. (λx. f(xx))(λx. f(xx)) recursion. Yt (λx. t(xx))(λx. t(xx)) t((λx. t(xx))(λx. t(xx))) = t(yt). Logic For Building Theories Practice-Based PoL: Amsterdam / 29

50 Remarks This calculus, encapsulated in one slide, is incredibly rich. A universal model of computation incomparably more wieldy than Turing machines. (Caveats: Church s thesis, resources). Indeed, in sugared form the basis of all contemporary functional programming languages (e.g. ML, Haskell). Kleene translated the basic results of recursion theory out of lambda calculus into the familiar φ n form. The untyped calculus allows e.g. terms like ω λx. xx self-application. Hence also Ω ωω, which diverges: Ω Ω Also, Y λf. (λx. f(xx))(λx. f(xx)) recursion. Yt (λx. t(xx))(λx. t(xx)) t((λx. t(xx))(λx. t(xx))) = t(yt). Historically, Curry extracted Y from an analysis of Russell s Paradox. Logic For Building Theories Practice-Based PoL: Amsterdam / 29

51 Types and Curry-Howard Correspondence Simple Type System for,. Variable Γ, x : t x : T Product Γ t : T Γ u : U Γ t, u : T U Γ v : T U Γ π 1 v : T Γ v : T U Γ π 2 v : U Function Γ, x : U t : T Γ λx. t : U T Γ t : U T Γ u : U Γ tu : T Logic For Building Theories Practice-Based PoL: Amsterdam / 29

52 Types and Curry-Howard Correspondence Natural Deduction System for, Identity Conjunction Γ, A A Id Γ A Γ B Γ A B -intro Γ A B Γ A -elim-1 Γ A B Γ B -elim-2 Implication Γ, A B Γ A B -intro Γ A B Γ A Γ B -elim Logic For Building Theories Practice-Based PoL: Amsterdam / 29

53 Types and Curry-Howard Correspondence If we equate they are the same! Logic For Building Theories Practice-Based PoL: Amsterdam / 29

54 Developments The Remarks Types and Curry-Howard Correspondence Developments Domain Theory Some Remarks and Questions for P-B PoL Logic For Building Theories Practice-Based PoL: Amsterdam / 29

55 Developments The stone the builders rejected... The Remarks Types and Curry-Howard Correspondence Developments Domain Theory Some Remarks and Questions for P-B PoL Logic For Building Theories Practice-Based PoL: Amsterdam / 29

56 Developments The Remarks Types and Curry-Howard Correspondence Developments Domain Theory Some Remarks and Questions for P-B PoL The stone the builders rejected... The and combinatory logic were a neglected corner of logic studied by a handful of people until Computer Science initially Strachey and Landin, with a part played by Roger Penrose put it centre stage in logical methods in CS. Logic For Building Theories Practice-Based PoL: Amsterdam / 29

57 Developments The Remarks Types and Curry-Howard Correspondence Developments Domain Theory Some Remarks and Questions for P-B PoL The stone the builders rejected... The and combinatory logic were a neglected corner of logic studied by a handful of people until Computer Science initially Strachey and Landin, with a part played by Roger Penrose put it centre stage in logical methods in CS. These calculi in turn put the study of substitution centre-stage not such a humble topic! Russell s paradox, cut-elimination, linearity and resources, decidability, complexity,... Logic For Building Theories Practice-Based PoL: Amsterdam / 29

58 Developments The Remarks Types and Curry-Howard Correspondence Developments Domain Theory Some Remarks and Questions for P-B PoL The stone the builders rejected... The and combinatory logic were a neglected corner of logic studied by a handful of people until Computer Science initially Strachey and Landin, with a part played by Roger Penrose put it centre stage in logical methods in CS. These calculi in turn put the study of substitution centre-stage not such a humble topic! Russell s paradox, cut-elimination, linearity and resources, decidability, complexity,... Paradoxes: not just biting the bullet not bugs but features! Recursion, fixpoints, the creative uses of computationally specified infinite objects. Logic For Building Theories Practice-Based PoL: Amsterdam / 29

59 Domain Theory Logic For Building Theories Practice-Based PoL: Amsterdam / 29

60 Domain Theory Providing extensional models for spaces satisfying led Dana Scott to Domain Theory. D = [D D] Logic For Building Theories Practice-Based PoL: Amsterdam / 29

61 Domain Theory Providing extensional models for spaces satisfying D = [D D] led Dana Scott to Domain Theory. Many interesting conceptual aspects of Domain theory: Logic For Building Theories Practice-Based PoL: Amsterdam / 29

62 Domain Theory Providing extensional models for spaces satisfying led Dana Scott to Domain Theory. D = [D D] Many interesting conceptual aspects of Domain theory: Reconciling paradoxes with fixpoints by introducing additional partially defined elements. Logic For Building Theories Practice-Based PoL: Amsterdam / 29

63 Domain Theory Providing extensional models for spaces satisfying led Dana Scott to Domain Theory. D = [D D] Many interesting conceptual aspects of Domain theory: Reconciling paradoxes with fixpoints by introducing additional partially defined elements. A general theory of partial information, dynamics of information increase. Logic For Building Theories Practice-Based PoL: Amsterdam / 29

64 Domain Theory Providing extensional models for spaces satisfying led Dana Scott to Domain Theory. D = [D D] Many interesting conceptual aspects of Domain theory: Reconciling paradoxes with fixpoints by introducing additional partially defined elements. A general theory of partial information, dynamics of information increase. Opens up the (analytical) topology of computation. Logic For Building Theories Practice-Based PoL: Amsterdam / 29

65 Domain Theory Providing extensional models for spaces satisfying led Dana Scott to Domain Theory. D = [D D] Many interesting conceptual aspects of Domain theory: Reconciling paradoxes with fixpoints by introducing additional partially defined elements. A general theory of partial information, dynamics of information increase. Opens up the (analytical) topology of computation. Conceptual ambiguity between ontic and epistemic interpretations. Logic For Building Theories Practice-Based PoL: Amsterdam / 29

66 Domain Theory Providing extensional models for spaces satisfying D = [D D] led Dana Scott to Domain Theory. Many interesting conceptual aspects of Domain theory: Reconciling paradoxes with fixpoints by introducing additional partially defined elements. A general theory of partial information, dynamics of information increase. Opens up the (analytical) topology of computation. Conceptual ambiguity between ontic and epistemic interpretations. A discussion of domain theory emphasizing conceptual aspects in my article in the Handbook of Philosophy of Information (ed. van Benthem and Adriaans, Elsevier 2008). Logic For Building Theories Practice-Based PoL: Amsterdam / 29

67 Some Remarks and Questions for P-B PoL The Remarks Types and Curry-Howard Correspondence Developments Domain Theory Some Remarks and Questions for P-B PoL Logic For Building Theories Practice-Based PoL: Amsterdam / 29

68 Some Remarks and Questions for P-B PoL The Remarks Types and Curry-Howard Correspondence Developments Domain Theory Some Remarks and Questions for P-B PoL The is essentially canonical for functional computation no 700 problem there. Logic For Building Theories Practice-Based PoL: Amsterdam / 29

69 Some Remarks and Questions for P-B PoL The Remarks Types and Curry-Howard Correspondence Developments Domain Theory Some Remarks and Questions for P-B PoL The is essentially canonical for functional computation no 700 problem there. What should Church s thesis for concurrency be? Logic For Building Theories Practice-Based PoL: Amsterdam / 29

70 Some Remarks and Questions for P-B PoL The Remarks Types and Curry-Howard Correspondence Developments Domain Theory Some Remarks and Questions for P-B PoL The is essentially canonical for functional computation no 700 problem there. What should Church s thesis for concurrency be? The gap between intension and extension: and its models vs. recursion theory. Applications of the recursion theory framework to partial evaluation and mixed computation, program specialization, computational learning theory, computer viruses! Logic For Building Theories Practice-Based PoL: Amsterdam / 29

71 Some Remarks and Questions for P-B PoL The Remarks Types and Curry-Howard Correspondence Developments Domain Theory Some Remarks and Questions for P-B PoL The is essentially canonical for functional computation no 700 problem there. What should Church s thesis for concurrency be? The gap between intension and extension: and its models vs. recursion theory. Applications of the recursion theory framework to partial evaluation and mixed computation, program specialization, computational learning theory, computer viruses! All based on mining the computation content of the S m n theorem and Kleene s Second Recursion Theorem. and its models are too extensional to allow access to this content. Can we find a unified theory? Logic For Building Theories Practice-Based PoL: Amsterdam / 29

72 Some Remarks and Questions for P-B PoL The Remarks Types and Curry-Howard Correspondence Developments Domain Theory Some Remarks and Questions for P-B PoL The is essentially canonical for functional computation no 700 problem there. What should Church s thesis for concurrency be? The gap between intension and extension: and its models vs. recursion theory. Applications of the recursion theory framework to partial evaluation and mixed computation, program specialization, computational learning theory, computer viruses! All based on mining the computation content of the S m n theorem and Kleene s Second Recursion Theorem. and its models are too extensional to allow access to this content. Can we find a unified theory? Game Semantics, full abstraction and full completeness. Again, see my article in the Handbook of Philosophy of Information (and, hopefully, forthcoming article in SEP). Logic For Building Theories Practice-Based PoL: Amsterdam / 29

73 Some Theses About Category Theory s Category Theory and Logic For Building Theories

74 Some Theses About Category Theory Logic For Building Theories Practice-Based PoL: Amsterdam / 29

75 Some Theses About Category Theory Category theory (and not just categorical logic) should be seen as part of logic or vice versa! Logic For Building Theories Practice-Based PoL: Amsterdam / 29

76 Some Theses About Category Theory Category theory (and not just categorical logic) should be seen as part of logic or vice versa! Logicians should learn category theory!! Logic For Building Theories Practice-Based PoL: Amsterdam / 29

77 Some Theses About Category Theory Category theory (and not just categorical logic) should be seen as part of logic or vice versa! Logicians should learn category theory!! Philosophers should learn category theory!!! Logic For Building Theories Practice-Based PoL: Amsterdam / 29

78 Some Theses About Category Theory Category theory (and not just categorical logic) should be seen as part of logic or vice versa! Logicians should learn category theory!! Philosophers should learn category theory!!! Category theory is the language of structure. It enables us to see common patterns far beyond what is otherwise possible. Logic For Building Theories Practice-Based PoL: Amsterdam / 29

79 Some Theses About Category Theory Category theory (and not just categorical logic) should be seen as part of logic or vice versa! Logicians should learn category theory!! Philosophers should learn category theory!!! Category theory is the language of structure. It enables us to see common patterns far beyond what is otherwise possible. Trivial example: isomorphism. Logic For Building Theories Practice-Based PoL: Amsterdam / 29

80 Some Theses About Category Theory Category theory (and not just categorical logic) should be seen as part of logic or vice versa! Logicians should learn category theory!! Philosophers should learn category theory!!! Category theory is the language of structure. It enables us to see common patterns far beyond what is otherwise possible. Trivial example: isomorphism. Category theory has a strong normative force: methodologically, it compels us to ask certain questions is it functorial, natural, universal? which point to the key notions in developing a theory. Logic For Building Theories Practice-Based PoL: Amsterdam / 29

81 Some Theses About Category Theory Category theory (and not just categorical logic) should be seen as part of logic or vice versa! Logicians should learn category theory!! Philosophers should learn category theory!!! Category theory is the language of structure. It enables us to see common patterns far beyond what is otherwise possible. Trivial example: isomorphism. Category theory has a strong normative force: methodologically, it compels us to ask certain questions is it functorial, natural, universal? which point to the key notions in developing a theory. Category theory enables us to think bigger thoughts. Many of the most interesting conceptual developments in modern mathematics cannot even be articulated without category theory. Logic For Building Theories Practice-Based PoL: Amsterdam / 29

82 Some Theses About Category Theory Category theory (and not just categorical logic) should be seen as part of logic or vice versa! Logicians should learn category theory!! Philosophers should learn category theory!!! Category theory is the language of structure. It enables us to see common patterns far beyond what is otherwise possible. Trivial example: isomorphism. Category theory has a strong normative force: methodologically, it compels us to ask certain questions is it functorial, natural, universal? which point to the key notions in developing a theory. Category theory enables us to think bigger thoughts. Many of the most interesting conceptual developments in modern mathematics cannot even be articulated without category theory. Examples: Cohomology, categorification, the microcosm principle. Logic For Building Theories Practice-Based PoL: Amsterdam / 29

83 s Some Theses About Category Theory s Category theory allows us to dualize our entire discussion of algebras to obtain a notion of coalgebras of an endofunctor. However, while algebras abstract a familiar set of notions (inductive data types, structural recursion), colagebras open up a new and rather unexpected territory, and provides an effective abstraction and mathematical theory for a central class of computational phenomena: Programming over infinite data structures: streams, lazy lists, infinite trees... A novel notion of coinduction Modelling state-based computations of all kinds The key notion of bisimulation equivalence between processes. Logic For Building Theories Practice-Based PoL: Amsterdam / 29

84 F -s Final F -coalgebras Labelled Transition Systems Transition Graphs as s The Final Some Remarks and Questions for P-B PoL Logic For Building Theories

85 F -s Let F : C C be a functor. An F -coalgebra is an arrow γ : A FA for some object A of C. We say that A is the carrier of the coalgebra, while γ is the behaviour map. An F -coalgebra homomorphism from γ : A FA to δ : B FB is an arrow h : A B such that F -s Final F -coalgebras Labelled Transition Systems Transition Graphs as s The Final Some Remarks and Questions for P-B PoL A h B γ FA Fh δ FB F -coalgebras and their homomorphisms form a category F Coalg. Logic For Building Theories Practice-Based PoL: Amsterdam / 29

86 Final F -coalgebras An F -coalgebra γ is final if for every F -coalgebra δ there is a unique homomorphism from δ to γ. Proposition 1 isomorphism. If a final F -coalgebra exists, it is unique up to F -s Final F -coalgebras Labelled Transition Systems Transition Graphs as s The Final Some Remarks and Questions for P-B PoL Proposition 2 (Lambek Lemma) isomorphism If γ : A FA is final, it is an Logic For Building Theories Practice-Based PoL: Amsterdam / 29

87 Labelled Transition Systems F -s Final F -coalgebras Labelled Transition Systems Transition Graphs as s The Final Some Remarks and Questions for P-B PoL Let A be a set of actions. A labelled transition system over A is a coalgebra for the functor Such a coalgebra LT A : Set Set :: X P f (A X). γ : S P f (A S) can be understood operationally as follows: S is a set of states For each state s S, γ(s) specifies the possible transitions from that state. We write s a s if (a, s ) γ(s). We think of such a transition as consisting of the system performing the action a, and changing state from s to s. Note that we regard actions as directly observable, while states are not. Logic For Building Theories Practice-Based PoL: Amsterdam / 29

88 Transition Graphs as s F -s Final F -coalgebras Labelled Transition Systems Transition Graphs as s The Final Some Remarks and Questions for P-B PoL Note that any labelled transition graph (or state machine ) with labels in A is a coalgebra for LT A. Examples 1. a b 1 2 This corresponds to the coalgebra ({1, 2}, γ) 2. c γ : 1 {(a, 1), (b, 2)}, 2 {(c, 2)} b a a c 1 {(b, 2), (c, 1)}, 2 {(a, 1), (a, 3)}, 3 Logic For Building Theories Practice-Based PoL: Amsterdam / 29

89 The Final The key point is this. Proposition 3 (Proc A, out). For any set A of actions, there is a final LT A -coalgebra F -s Final F -coalgebras Labelled Transition Systems Transition Graphs as s The Final Some Remarks and Questions for P-B PoL Logic For Building Theories Practice-Based PoL: Amsterdam / 29

90 The Final F -s Final F -coalgebras Labelled Transition Systems Transition Graphs as s The Final Some Remarks and Questions for P-B PoL The key point is this. Proposition 3 (Proc A, out). For any set A of actions, there is a final LT A -coalgebra We think of elements of the final coalgebra as processes. The final coalgebra provides a universal semantics for transition systems, which is fully abstract with respect to observable behaviour. Logic For Building Theories Practice-Based PoL: Amsterdam / 29

91 The Final F -s Final F -coalgebras Labelled Transition Systems Transition Graphs as s The Final Some Remarks and Questions for P-B PoL The key point is this. Proposition 3 (Proc A, out). For any set A of actions, there is a final LT A -coalgebra We think of elements of the final coalgebra as processes. The final coalgebra provides a universal semantics for transition systems, which is fully abstract with respect to observable behaviour. All of this generalizes to a large class of endofunctors. Logic For Building Theories Practice-Based PoL: Amsterdam / 29

92 Some Remarks and Questions for P-B PoL F -s Final F -coalgebras Labelled Transition Systems Transition Graphs as s The Final Some Remarks and Questions for P-B PoL Logic For Building Theories Practice-Based PoL: Amsterdam / 29

93 Some Remarks and Questions for P-B PoL s naturally model state-based systems. They provide a promising basis for reconciling ontic and epistemic views of states. The final coalgebra is a universal solution hence unique up to isomorphism to the problem of constructing states as determined purely by their observational behaviour. F -s Final F -coalgebras Labelled Transition Systems Transition Graphs as s The Final Some Remarks and Questions for P-B PoL Logic For Building Theories Practice-Based PoL: Amsterdam / 29

94 Some Remarks and Questions for P-B PoL F -s Final F -coalgebras Labelled Transition Systems Transition Graphs as s The Final Some Remarks and Questions for P-B PoL s naturally model state-based systems. They provide a promising basis for reconciling ontic and epistemic views of states. The final coalgebra is a universal solution hence unique up to isomorphism to the problem of constructing states as determined purely by their observational behaviour. ic logic. A generalized modal logic which can be read off systematically from the type functor T. Generalized duality theory. Logic For Building Theories Practice-Based PoL: Amsterdam / 29

95 Some Remarks and Questions for P-B PoL F -s Final F -coalgebras Labelled Transition Systems Transition Graphs as s The Final Some Remarks and Questions for P-B PoL s naturally model state-based systems. They provide a promising basis for reconciling ontic and epistemic views of states. The final coalgebra is a universal solution hence unique up to isomorphism to the problem of constructing states as determined purely by their observational behaviour. ic logic. A generalized modal logic which can be read off systematically from the type functor T. Generalized duality theory. Corecursion, coinduction: mathematically well-founded treatment of non-well-founded objects. Logic For Building Theories Practice-Based PoL: Amsterdam / 29

96 Some Remarks and Questions for P-B PoL F -s Final F -coalgebras Labelled Transition Systems Transition Graphs as s The Final Some Remarks and Questions for P-B PoL s naturally model state-based systems. They provide a promising basis for reconciling ontic and epistemic views of states. The final coalgebra is a universal solution hence unique up to isomorphism to the problem of constructing states as determined purely by their observational behaviour. ic logic. A generalized modal logic which can be read off systematically from the type functor T. Generalized duality theory. Corecursion, coinduction: mathematically well-founded treatment of non-well-founded objects. Examples: non-well-founded sets, even non-well-found proofs! Logic For Building Theories Practice-Based PoL: Amsterdam / 29

97 Logic As A Tool For Building Theories Some Challenges for Practice-Based Philosophy of Logic Logic For Building Theories

98 Logic As A Tool For Building Theories Logic As A Tool For Building Theories Some Challenges for Practice-Based Philosophy of Logic Logic For Building Theories Practice-Based PoL: Amsterdam / 29

99 Logic As A Tool For Building Theories Logic As A Tool For Building Theories Some Challenges for Practice-Based Philosophy of Logic Computer Science theories of: Processes of various kinds, how to mathematically describe and reason about them. Information: statics of information representation, dynamics of information flow. Logic For Building Theories Practice-Based PoL: Amsterdam / 29

100 Logic As A Tool For Building Theories Logic As A Tool For Building Theories Some Challenges for Practice-Based Philosophy of Logic Computer Science theories of: Processes of various kinds, how to mathematically describe and reason about them. Information: statics of information representation, dynamics of information flow. The formulation and development of these theories uses a lot of logic essentially is logic, broadly (and properly) construed. Logic For Building Theories Practice-Based PoL: Amsterdam / 29

101 Logic As A Tool For Building Theories Logic As A Tool For Building Theories Some Challenges for Practice-Based Philosophy of Logic Computer Science theories of: Processes of various kinds, how to mathematically describe and reason about them. Information: statics of information representation, dynamics of information flow. The formulation and development of these theories uses a lot of logic essentially is logic, broadly (and properly) construed. Logic in the mode of open-ended, outward-reaching modelling, rather than conservative codification. Logic For Building Theories Practice-Based PoL: Amsterdam / 29

102 Logic As A Tool For Building Theories Logic As A Tool For Building Theories Some Challenges for Practice-Based Philosophy of Logic Computer Science theories of: Processes of various kinds, how to mathematically describe and reason about them. Information: statics of information representation, dynamics of information flow. The formulation and development of these theories uses a lot of logic essentially is logic, broadly (and properly) construed. Logic in the mode of open-ended, outward-reaching modelling, rather than conservative codification. Considerable potential beyond Computer Science: in physics, biology, cognitive and social sciences etc. Logic For Building Theories Practice-Based PoL: Amsterdam / 29

103 Some Challenges for Practice-Based Philosophy of Logic Logic As A Tool For Building Theories Some Challenges for Practice-Based Philosophy of Logic Logic For Building Theories Practice-Based PoL: Amsterdam / 29

104 Some Challenges for Practice-Based Philosophy of Logic Analyze a real, contemporary research programme in mathematics, logic or theoretical computer science. Logic As A Tool For Building Theories Some Challenges for Practice-Based Philosophy of Logic Logic For Building Theories Practice-Based PoL: Amsterdam / 29

105 Some Challenges for Practice-Based Philosophy of Logic Logic As A Tool For Building Theories Some Challenges for Practice-Based Philosophy of Logic Analyze a real, contemporary research programme in mathematics, logic or theoretical computer science. Study the choices made, the reasons given, the methodological disagreements, what these were really about, why certain contributions were decisive, why conceptual arguments about approaches were decided in a certain way. Logic For Building Theories Practice-Based PoL: Amsterdam / 29

106 Some Challenges for Practice-Based Philosophy of Logic Logic As A Tool For Building Theories Some Challenges for Practice-Based Philosophy of Logic Analyze a real, contemporary research programme in mathematics, logic or theoretical computer science. Study the choices made, the reasons given, the methodological disagreements, what these were really about, why certain contributions were decisive, why conceptual arguments about approaches were decided in a certain way. This will engage the interest, enthusiasm, and eventually the active participation of the practitioner community. Logic For Building Theories Practice-Based PoL: Amsterdam / 29

107 Some Challenges for Practice-Based Philosophy of Logic Logic As A Tool For Building Theories Some Challenges for Practice-Based Philosophy of Logic Analyze a real, contemporary research programme in mathematics, logic or theoretical computer science. Study the choices made, the reasons given, the methodological disagreements, what these were really about, why certain contributions were decisive, why conceptual arguments about approaches were decided in a certain way. This will engage the interest, enthusiasm, and eventually the active participation of the practitioner community. Contrast: Philosophy of Physics vs. Philosophy of Logic and Mathematics. Logic For Building Theories Practice-Based PoL: Amsterdam / 29